COVALENCY EFFECTS IN MAGNETIC INTERACTIONS

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COVALENCY EFFECTS IN MAGNETIC

INTERACTIONS

B. Tofield

To cite this version:

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CO

VAL ENC Y EFFECTS AND MA GNETIC

H

YPEISFINE' INTERA C TIONS.

COVALENCY EFFECTS IN MAGNETIC INTERACTIONS

B. C. TOFIELD

Materials Physics Division, A. E. R. E., Harwell, OX11 ORA, U. K.

RBsumB.

-

On prksente une revue des interactions magnetiques dans le but de dkgager les informations sur la covalence qui peuvent en Stre dkduites.

Les interactions hyperfines sur des ligands, les champs hyperfins dans les mktaux mesurks par spectromktrie Mossbauer et diffracti~n neutronique sont prksentkes en detail, en insistant sur la dernihre approche qui est la plus productive en ce domaine. Les informations dkj& recueillies sont confrontkes aux connaissances en chimie et on essaie de degager les axes intkressants pour de futures etudes.

Abstract. - Magnetic interactions, categorised into ligand based, metal based and neutron diffraction, are reviewed with regard to the information they give on covalency. The ligand hyperfine interaction, the metal hyperfine field measured by the Mossbauer effect and neutron diffraction measurements are mentioned in some detail, with particular emphasis on the latter as being the most productive experimental technique in this field. The information already gathered is assessed with reference to chemical knowledge, and useful avenues for future investigation are suggested.

1. Introduction. - The ionic model occupies a central position, both in the teaching of solid state inorganic chemistry and in the explanation of many solid state phenomena in the current literature. I t has been used for example to rationalise many thermodynamic data and in this context has been lucidly described in text books such as those of Phillips and Williams [I] and Johnson [2]. Such descriptions rely on the concept of the ionic radius, more or less transfe- rable from compound to compound, and the derivation of accurate radii and the determination of the effect on them of factors such as ionic spin, coordination number, bond distortion and so on, has received new impetus with the publication of many accurate crystallographic data in the last few years. The tables of Shannon and Prewitt [3] represented a significant milestone in this endeavour, and stimulated much supplementary activity, detailed in the revised list of Shannon [4]. Such work is inherently empirical in nature, but even in some of the most sophisticated quantitative lattice potential calculations currently performed, to explain [ 5 ] , for example the occurence of very short fluorine-fluorine distances in anion-excess fluorites such as Cal -,YxF2

+,,

it has not been found necessary to invoke deviations from formal ionic charges.

Nevertheless, there are many indications in thermodynamic and structural solid state chemistry of deviations from ionic model behaviour, even though self- compensating features of the ionic model do tend to compensate for departures from formal ionic charges. This is discussed by Phillips and Williams [I], who detail the covalent contribution to heats of formation of

B-group metal compounds by comparison with the compounds formed by A-metal ions of similar radii. More traditional evidence has been the reduction of coordination numbers associated with metals suspected of showing significant covalency, or directed bonding, in their compounds, particularly for B-group metals such as silver, zinc, mercury, and so on. The elastic constants of tetrahedrally bonded materials with zinc blende or wurtzite structure have been related to the degree of covalency by Phillips [6] who showed how the ratio of bending to stretching force constants increased linearly with decrease of ionicity, as determined from the dielectric properties of the crystals. The shear constants reflect the magnitude of covalent bond charge and resistance to bond bending, but unfortunately such data are either not available, or not so straight- forwardly analysed in most situations of interest to the inorganic chemist, where either the symmetry is too low, the unit cell too complicated or the ionicity higher than in the simple tetrahedral semiconductors.

The adoption of structures, particularly layer lattices, not explicable for any combinations of ionic radii is also generally accepted to be evidence for metal-ligand covalency, as is the shrinkage [I, 4, 71 of effective ionic radii for some metals such as Cu', Tli, Pb2+ on bonding to relatively more polarisable ligands (e. g. I- c. f. F- or

S2-

C. f. 0'-) this effect being intimately related to the thermodynamic features mentioned. However, it is not at all clear what is the amount of covalency relative to other polarization effects such as van de Waals interactions which is necessary to produce such structural manifestations. Although layer lattice compounds such as those of the

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C6-540 B. C . TOFIELD

transition metal dichalcogenides have been much studied lately with regard to the intercalation of metals, e. g. lithium into titanium disulphide [a], and organics, e. g. pyridine into tantalum disulphide [9], there do not seem to be any well established rules for predicting what reduction of the formal charge of the anion by covalency is required before the van der Waals attraction between layers of sulphur, chlorine, or other anions often found in layer lattices, becomes the dominant structure-determining feature. Tetrahedral coordination of the metal atoms, a consequence if covalent interactions are strong, according to Phil- lips [6], is not a feature of many layer lattice compounds. From structural chemistry also one might enquire about the role and the nature of covalency in the defect behaviour of high oxidation state oxides. Does the occurrence of crystallographic shear [lo], and the elimination of point defects in oxides of Ti, Nb and W, for example, depend at least in part on relatively strong ligand-to-metal n-bonding, permitting edge-sharing or face-sharing of octahedra ? Shear is not observed in isovalent oxides of the post-transition B-group metals such as Sn and Sb although for these, covalency is surely just as important, but is now dominated by ligand- to metal s-bonding. The structures of A + B S + 0 3 compounds have been justified in similar terms [ll]. Some years ago I suggested it would be interesting to study BaPtO, to compare with related phases for earlier transition metals. Now PtlV, d6 low spin, is strongly stabilised by o-covalent interaction in an octahedral environment but BaPtO, does not form as a perovskite. Phases such as Ba4Pt06 are much more easily prepared however and these facts seem to be another structural consequence of strong covalency :

in Ba,Pt06 each oxygen linked to Pt is shared only with a barium ion, but in BaPtO, the

P~oZ-

octahedra are not isolated and we can say that the oxygen is too strongly electronegative to be able to donate sufficient electron density to satisfy two pt4+ ions. In other words ~ t 0 : - is now effectively an oxy-anion like COO:-,

F~o:-

and other high oxidation state metal-oxygen solid state complexes. It is to provide explanations to features such as these and also to be able to quantify predictive chemistry that the chemist is interested in covalency. In solid state inorganic chemistry and physics this seems to be most easily comprehended from the standpoint of the ionic model with effective charges of ions reduced by some amount from the formal ionic charges. This concept is nicely quantified in the molecular orbital model of covalency, applied below to the case of transition metal ions.

One of the most fertile areas for illustrating effects of covalency has indeed been the solid state compounds of the transition elements [12]. The wide variety of size and charge of ion possible in structure types such as perovskite permits many different regimes of electrical and magnetic behaviour, some of which demand an explanation involving covalency effects. Because many

of these ions have a nonzero spin, associated more or less with a specific d or f orbital, the fact that a covalent interaction between a closed shell anion orbital and an unpaired spin on a metal ion alters the spatial distribution of this spin provides a sensitive probe for investigating at least a part of the total bonding interaction. It is for this reason that most quantitative, or semi- quantitative data on covalent effects have been obtained by measurement of magnetic interactions, principally by magnetic resonance, Mossbauer spectroscopic or neutron diffraction techniques.

The direct measurement of unpaired spin distribution, either by examining the region near the ligand nucleus, transferred by covalent interactions, using Iigand hyperfine interactions (LHFI), or the total spin distribution by neutron diffraction seems to provide the most direct information. Hyperfine interactions with the metal ion nucleus measured either by ESR or Mossbauer spectroscopy are clearly also very sensitive to d electron covalency, but being modulated by s orbitals, are sometimes more difficult to interpret quantitatively. This review will attempt to assess and compare the information gathered by the various techniques. Many trends are clearly observable ; these are indeed often thought to be satisfactory if agreeing with the type of thermochemical data mentioned. Nevertheless there are inconsistencies, both compared with thermochemical data and in comparing measurements of one technique with another, although sometimes these are understandable in that different techniques probe different regions of the electron charge cloud. One quickly realises the need for theoretical calculation to supplement the experimental effort, both to interpret the data obtained and to draw attention to defects in the models used for interpretation, although there is much virtue in simple models, especially when used for comparative purposes, unless hopelessly physically unrealistic. Several calculations have seemed to indicate that the molecular orbital model of covalency in transition metal complexes, unlike the ionic crystal field model, might be acceptable on these terms.

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COVALENCY EFFECTS IN MAGNETIC INTERACTIONS C6-541

and an aim of theoretical calculations should be to synthesise the various pieces of data gained on particular compounds. Also, this review is competent to deal only with quite simple lattice-type compounds. All the techniques mentioned, with the exception of neutron diffraction, have been widely applied to bonding in complexes which in many respects differ only from simpler compounds in the bulkier nature of the ligand. As the difficulties presently encountered even in understanding quite simple solids become more understood the emphasis generally will naturally shift to more complicated chemical situations involving complexes, often trying to simulate biological situations. Although most of the presently obtained knowledge on complexes will not be much mentioned here I will hope to indicate some situations where experiments might usefully be carried out in the near future.

This review will also show an emphasis towards neutron diffraction methods, because of the author's greater familiarity with this area, because it is the least widely appreciated and because, by virture of spin density mapping, it has the greatest promise for provid- ing useful new data. Although I am no longer actively involved in this field I would like to use this occasion to encourage others to perform accurate spin density measurements using the excellent facilities which now exist. The task has hardly begun but the rewards would be great.

A review of the progress to date of the covalency information gained from neutron diffraction was written two years ago [27]. One can justify a further

discussion because new data have been obtained both by neutron diffraction and other techniques, because there is the chance to mention the correlations between magnetic properties and other parameters recently drawn [14], and because there is the possibility of proposing some directions which future study might usefully take.

2. Experimental Data Interpretation. - The expe- rimenter obtains numbers which he must relate to the physical system under investigation. The extraction of covalency information from magnetic measurements has generally been achieved using a simple molecular orbital model. This model was first invoked [15] in this context to account for the discovery of a hyperfine interaction with the chlorine nuclei in 1r~12- and the reduced g-factor in this complex, and it is rather appo- site that the spin density in this complex revealing the presence and symmetry of the spin covalently delocalis- ed onto the chlorine atoms in K,IrCI, has recently been determined. This complex is rather unusual in fact, being based on a third-row transition metal ion bonded to chlorine ; most of the data collected by spin resonance or neutron diffraction methods have been on first row metals' bonded to oxygen or fluorine. Hopefully this situation will change before too long, but we must note that Mossbauer spectroscopy has recently provided some interesting information on covalency involving iodine ligands.

If the positive and negative axes of an octahedron for the x, y and z directions are labelled 1 and 4, 2 and 5, and 3 and 6 respectively, the antibonding orbitals for d-electron bonding are :

1

+3,2-2. = N, (d3.2-.2 -

-

,I,(- 2pr3

+

2pz6

+

pxt

-

px4

+

py2

-

py5)

-

Jiz

where

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C6-542 B. C . TOFIELD It should be remembered that these wavefunctions

describe the antibonding orbitals derived mainly from d electron states in a transition metal complex. If there are no outer d electrons there cgn still be significant covalency in compounds of (formally) ~ i ~ " , Nb5

",

Mo6+ and so on. This will also be the case for o bonding in octahedral compounds with no e, electrons (e. g. Cr3 '(t:,), 1r4+(tZg)). Magnetic measurements will not be sensitive to covalency in these situations, but charge sensitive measurements (e. g. quadrupole coupling constants) will be. Similar remarks apply to diamagnetic complexes containing antibonding d-electrons in, for example, low spin d6 octahedral, or square- planar d8 configuration. Nor will magnetic measurements reflect bonding into other than d (or f) states on the metal, except sometimes indirectly. Yet ions such as Cd2+ and z n 2 + with a completed d-shell seem to be as covalent in their behaviour as Ni2" and Mn2+ where d-electron covalency is well established by magnetic measurements. The bonding to outer s and p levels occurring for the B-group metals will occur in transition compounds as well, although to a lesser extent because of the lower effective nuclear large. The Mossbauer isomer shift is very sensitive to the partial occupation of outer s orbitals. Sometimes, in favourable situations, the experimental data can directly reveal deficiencies in the model. This is the case for spin polarised e, electrons in c r 3 + compounds where the model given above predicts no o-bonding effect to be revealed in the magnetic behaviour. For Mn2', on the other hand, where both o and n bonding are measured by neutron diffraction or spin resonance, the consequences of spin polarisation cannot be readily disen- tangled [16].

In general, therefore, although one can usefully use the covalency parameters obtained from experiment via the simple MO model to demonstrate trends in a series of compounds, one must be cautious when seeking a true physical interpretation. It is the province of the theoretician to be able to reproduce by calculation the numbers obtained experimentally, and to inform us, both on the approximations involved on the way and their likely consequences, and on the real physical significance of the numbers, i. e. where the models are good or bad and where they need modifying. Such interaction is essential.

The admixture coefficients I,, I, and I,, which may be derived from experiment contain both covalency and overlap information. If y is the admixture parameter in a bonding orbital, then to first order

Although normally neglected in qualitative discussions, the treatment of I as purely a measure of bonding may be a rather serious approximation in situations where the metal-ligand admixture and overlap are quite strong. The rather small s admixture parameter observed for transition metal complexes may arise principally from overlap effects.

The complex as proposed is presumed isolated from other magnetic complexes, as occurs in the experimental situation for EPR experiments. It is customery to refer to the fraction of unpaired spin (f) transferred to a single ligand orbital when the metal d orbitals of the appropriate symmetry are singly occupied (d3, d5 high spin for t,, .n electrons, d5 and d8 for ep o electrons). Then

~ b 2

62 Nb2 2

~ , 2

fu =

,

fs =

-

2, and

f

= - A' (10)

3

,

4 n '

The relations are not correct for other d states and, in reading the literature one has both to ascertain what symbols are being used and whether f really means f

or is used as a measure of 1 in eq. (10) in all cases. The simple LCAO model assumes that the combining atomic orbital states are radially unchanged. However, the charge transfer occurring will undoubtedly cause modifications of screening effects and for transition metal complexes this is expected to allow an increase in the radial extent of the d orbitals. Such an effect may occur via transfer of charge to outer s and p orbitals also. On the other hand, the Coulombic effect of the other ions in the lattice may cause either an expansion or a contraction. Jorgensen [17] distinguished the expansion of d orbitals via screening, or central field covalency, from the charge transfer occurring via the LCAO molecular orbital model, or symmetry restricted covalency, to account for the nephelauxetic effect. It would appear that these mechanisms are sometimes confused in the literature, but anyway it is not at all clear whether the MO model described is sufficiently close to physical reality to allow such fine distinctions to be meaningful. One must appeal again to calculation. Experimentally, the measurement of neutron form factors may throw light on the radial functions involved, but here the evidence so far is equivocal, which encourages caution in attempts to explain other data such as reduced quadrupole coupling constants on the basis of significantly expanded d functions.

3. The Ligand interactions. - Writing the antibonding orbitals in the form

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COVALENCY EFFECTS I N MAGNETIC INTERACTIONS C6-543

+

2 & A s Xzs Xzpa ( 1 3 ~ )

I

+,(TI

l2

= dn2(1

-

22)

+

( 1 4 ~ )

+ 2(Az S, d;

-

A, dz x2pn) (14b)

+ 1 2

~ g p n

.

(14~)

The spin density can be considered to be in three parts, a term of the form dZ(l

-

A'), a term of the form A2

x2

and a term containing overlap integrals and products of metal and ligand wavefunctions. The magnitude of the spin density on the ligand is directly proportional to the admixture parameter and thus provides a very direct measurement of the covalency in the complex. The perturbation of the ligand nuclear levels caused by this new magnetic field, the ligand hyperfine interaction LHFI, provided the first direct demonstration of covalency in transition metal complexes [IS], and has since been measured in many different complexes by spin resonance techniques. Recently, measurement of the hyperfine field at '''1 by Mossbauer spectroscopy has also provided covalency information, for example for CrI, [19].

The Hamiltonian for the interaction at the ligand may be written generally as

X = - g N ~ l u H s I (1 5 ) where H is the transferred hyperfine field and I the nuclear moment on the ligand. For 19F, I =

3

and many spin resonance data have been obtained on fluoride complexes. For 160, I = 0 and spin resonance data have only been collected on oxides with the fairly recent preparation of materials doped with 170 (I = +).

Where I

>

+

(170, the heavier halides) there will be a quadrupole interaction in addition. Thus to fit the lz9I Mossbauer spectra of CrI, below the Curie temperature the Hamiltonian needed was

where eq is the electric field gradient along the principal

(2) axis of the EFG tensor (e is the charge on the

proton), Q is the nuclear quadrupole moment and y

is the asymmetry parameter. Such an analysis yields H, eq and the angles between the coordinate systems of the EFG and the hyperfine field. The results may be obtained from polycrystalline samples in suitable cases which offers some advantages in comparison with spin resonance results where the hyperfine interaction must be measured as a function of orientation withsingle crystals.

y = 0 in the case that the symmetry is axial, as along a metal-ligand bond in an octahedral complex, the frequently encountered geometry in spin resonance measurements. The spin Hamiltonian used to describe the ligand interaction in magnetic resonance measurements is then conveniently expressed in the form :

+

Bf(S, I,

+

S, I,)

where the first term is the nuclear Zeeman interaction with the applied field. When the ground state of the magnetic ion is an orbital singlet A' and B' may be quite directly related to the contributions from the separate types of orbitals containing unpaired spin, and the dipolar contribution, Ad from the metal unpaired electrons :

A' = As

+

2(Ad

+

A, - An) (18) B' = A, - (A, f A,

-

A,) (19) where

and

R is the metal-ligand distance, and the dipolar and contact hyperfine interactions corresponding to a singly occupied p orbital, and s orbital, respectively are :

There is also an isotropic contact term arising from the unpaired ligand p orbitals due to polarisation of core s electrons, which is of the form

where k = 2 g , p, p,

<

r i 3

>

u and K 0.1 [20]. A: is normally much smaller than A, but may be important where, to first order,f, = 0 by symmetry, as in Cr3+, d3. However, calculations to estimate the magnitude of K have only been done for F- ions so there is considerable uncertainty as to the correct values to use for heavier ligands such as I-.

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C6-544 B. C. TOFIELD

could be determined independently for CrI, [19]. Such measurements were comprehensively discussed by Owen and Thornley [20], and as they note, the data may be interpreted at various levels of approximation. Certainly the ligand wavefunctions must be known to provide

I

$(0)

l2

and

<

r i 3

>

and these will vary with the charge state which will vary with the covalency. This could make a difference of up to 10

%

for F - ;

for 0'-, and for more complicated ligands uncertain- ties of at least this magnitude are probably inherent for any chosen charge state. The real ionic charges corrected for covalency must be known to permit the correct determination of A, and the R must also be known. For EPR or ENDOR work, this may well be subject to some uncertainty as the magnetic complex is doped into a diamagnetic host. Other problems can arise from competing processes not accounted for in the simple MO model. These can be quite important, but difficult to estimate quantitatively and include contributions from inner ligand electrons (e. g. 2p electrons for chlorine [21]), admixture with empty ligand orbits (expected to be particularly important in cases where metal-to-ligand z bonding is occurring), spin polarised transfer to empty metal orbitals (especially noticeable for d3 ions and discussed beIow), and spin-orbit admixture of excited charge-transfer states into the ground state [20].

Once again, we see that a quantitative interpretation requires extensive calculation, but that qualitative interpretation of data, and of trends via the MO model is in order. This is particularly important as comparison can then be made with data collected by other techniques such as neutron diffraction and Mossbauer spectroscopy. Although the collection of magnetic resonance LHFI data is an active field still, the complexity of the interpretation for more complex materials, which includes all complexes with poly- atomic ligands, sometimes makes the extraction of quantitative covalency information quite difficult. Fairly recent discussions have been given, for example, for CuCli- and C u ~ r i - (ref. [22]), for proton NMR in RU(NH,):+ (ref. 1231) and for 170 LHFI in 6 3 ~ ~ ( ~ 2 0 ) 2 6 ' (ref. [24]). The determination of spin density distributions by neutron diffraction will be as important an advance in this area as was the measurement of covalency parameters by neutron diffraction for simpIe oxides, and fluorides.

The quadrupole interaction is quite complementary to the magnetic hyperfine interaction as it reflects not spin but charge transfer. eq and possibly q can be extracted from the parameters of the spin Hamiltonian. In a crystal,

The Sternheimer antishielding and shielding parameters y, and R [0

<

(1

-

R) < 11 describe the induced distortions of the charge cloud of the ligand due to the lattice and due to the valence electrons. R is

quite small (0.2-0.3 often), y, is large and negative (often

-

lo), and neither is known very precisely. The lattice contribution has often been neglected in interpretation of quadrupole coupling constants as it is often considerably smaller than the valence term. For ligand p electrons in an octahedral complex the valence contributions to q (neglecting the shielding factor) are

4

q = - - < r - 3

5

>

for p, and

If then, we consider bonding to metal d states only, and that A = y (eq. (9)) then a quadrupole interaction arises when there is a net imbalance of electrons (N) in the p, relative to the px and p, orbitals [25]. If

then

fQ may be related tof, and

f,

[20, 261 and the relations have been tabulated [I 3,271.

In general, fQ will measure different combinations of f, and f, (which are fixed relations of A, and

A,

by eq. (10)) than obtained through the ligand hyperfine interaction although for high spin d5 ions (Mn2+, Fe3+) fQ =

f,

- f, and for d8 complexes fQ =

fo.

fQ, relecting charge transfer also provides results on non-magnetic ions and is, for example, 2(f,

-

f,) and 2

f,

for do and low-spin d6 species respectively.

Because of the several approximations involved in this approach, however, any absolute interpretation of

fa

is highly qualitative, although broad consistency with LHFI-measured spin transfer coefficients is claimed [28, 291 for Mn2+, Fe3

+,

Cr3' and Ni2+ in 170-doped MgO.

A more promising approach is possible where the observed eq may be normalised with reference to an atomic species. This is possible for C1, Br and I and

in this case, at least if one is considering the same formal charge as in the simple halides, the

<

r - 3

>

and (1

-

R)

terms will disappear if we define

eq (crystal)

'

Q = eq (atomic)

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COVALENCY EFFECTS IN MAGNETIC INTERACTIONS C6-545 A fair range of compounds with the K,IrC16

structure (Fig. 1) have been studied. Here the halide ion is indeed bonded to one metal atom only, as part of an octahedral complex, and the simple relations o f f , and f, to fQ, mentioned above, are appropriate.

FIG. 1. -The crystal structure of KzIrC16. The only variable positional parameter is that of C1 which may be slightly displaced from (00

a)

and equivalent positions to (00z). The structure is related to that of an ordered perovskite K2Ir Cl6 where is a vacancy so that the 1rCIt- octahedra are isolated from one another. Alternatively it may be viewed as an antifluorite Kz(IrC16).

It would appear that bromides are consistently more covalent than chlorides but fQ is generally very similar for bromides and iodides. This is interesting as other data generally suggest that the reverse if anything might be true (chlorides

-

bromides and less covalent than iodides). Whether this,reflects neglect of essential factors, such as overlap effects, in the model, or different mechanisms of bonding does not seem to be clear at the present time. Other trends apparent are a similarity of 4d and 5d covalency for the chlorides and bromides of Pd and Pt, and a greater covalency per ligand for tetrahedrally coordinated compared to octahedrally coordinated snIV and PbIV. This latter result appears to be supported by '19Sn Moss- bzuer isomer shift data [30], apparently indicating also, therefore, a greater total charge transfer in the tetrahedral case. It is interesting to compare these results with the fairly similar conclusions (see below) reached from neutron diffraction experiments on oxides containing octahedrally and tetrahedrally coordinated rnxtal ions, although for the B-group metals the bonding is dominantly by charge transfer to metal outer s and p orbitals and tetrahedral coordination encourages favourable sp3 hybridisation. For the transition metals this mechanism is less important as the 4s and 4p orbitals are higher in energy and radially more diffuse.

Mossbauer data for Fe3' oxides indicate significant 4s population ( 2 0.1 electron) as discussed below and there is also NQR evidence pointing to 4s and 4p involvement in bonding, in particular the C1 quadrupole coupling constant of KMnCl, where for Mn2+ we recall that fQ should equal

f,

-

f,.

However for a linear Mn-C1-Mn situation,

f,

-

f,

by NMR is roughly zero but fQ is 16.7

%

[31] (for CsMgCl,, fQ is 8.1

%).

Finally, by comparing the fQ data for K,MC16 (M = W to Pt) it appears that covalency must increase, as expected, across the 5d series.

Most halides of divalent and trivalent metals do not have the simple geometry considered above and a proper analysis invoking the particular geometry must be made if a comparison of quadrupole data with spin transfer data is required. This has been done [19] recently using Mossbauer data for CrI, which is discussed below. Because iodine is a Moss- bauer nucleus it is however appropriate to mention here the qualitative picture of bonding for iodine compounds which can be presented by combining isomer shift with quadrupole coupling data [32].

A semi-empirical relation relates the isomer shift

relative to ZnTe to the 5s and 5p occupations :

6 = ah, -i- bh,

+

6, (28) where h, is the number of 5s holes and h, is the number of 5p holes. a and b will be of opposite sign as the loss of 5p electrons reduces the 5s screening, and de Waard estimated [32] the best values from various compilations with the probable errors to be,

and So =

-

0.54

2

0.02 (mm s-I). We recall (eqs (25), (26)) that the sign of q will reflect the dominance of TC or B bonding (negative and positive respectively)

and thus if 6 - 6, is plotted against q an indication of the nature of the bonding is displayed (Fig. 2). The straight lines represent bonds with predominant

n or a bonding and little hybridisation. For positive 6

-

6,, points between the lines represent mixed

TC and a bonding and points below indicate 5s involvement in the bonding. It would be encouraging to see the broad features of such a plot reproduced by calculation. For the group valence, KIO,, IF,, etc. a significant 5s contribution is indicated.

We have mentioned the outlines for the interpretation of the LHFI only for the case of transition metal ions with orbital singlet ground states. When the ground state is orbitally degenerate spin orbit coupling acts in first order to scramble orbital and spin states which makes the extraction of covalency information considerably more tedious 1331. In general it is not straightforward to extract

f,

and

f,

individually and either approximations must be made (e. g.

f,

=

f,)

or the ligand hyperfine information must be combined with other information such

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FIG. 2. - A plot of isomer shift versus q for 1291 compounds.

The straight lines indicate pure a and pure a bonding according

to the simple Townes-Dailey model. For positive 6

-

60 points

between the lines represent mixed a and a bonding, and negative 6 - 60 indicates 5s involvement in the bonding (from ref. [32]).

as the g-factor reduction. The magnitude of the interaction may, however, be related to the total bonding interaction so that comparisons of this sort for different ligands and a given metal ion have been made in a few cases. Nevertheless, the great majority of the quantitative information gained so far has been collected on spin only metal ions and except in special cases we will not discuss in any detail spin distribution measurements on systems where the spin-orbit coupling is not quenched to first order. For Co2 + (d7)

and Ir4+ (d5 low spin) quite comprehensive discussions have been given [34,35].

A similar situation applies to the neutron diffraction case for transition metal species. The treatment for magnetic neutron scattering of combined orbital and spin moment in the most suitable fashion remains a topic of lively theoretical interest (see for example ref. [36] and references therein), but because the effects of covalency are generally small in the first instance any added uncertainty due to unquenched orbital moment makes extraction of bonding parameters of any significance very difficult indeed, especially as the effects may be quite similar in some cases [37].

4. Metal based interactions. - Several parameters primarily related to metal electron charge or spin distributions are sensitive to covalency, although often in a fairly complex way. These include hyperfine interactions measurable by ESR or Mossbauer spectroscopy, supertransferred hyperfine interactions, metal isomer shifts and quadrupole splittings obser-

vable in Mossbauer work, particularly of 5 7 ~ e , orbital reduction factors of orbitally degenerate ground state species accessible by ESR or magnetic susceptibility, g-values measurable by ESR where spin-orbit coupling acts in second order, core level splittings observed by photoelectron spectroscopy, and ligand field splittings and electron repulsion parameters measured in electronic spectra.

A reduction of the orbital contribution to the

magnetic moment was observed for the complex 1rC12- by Owen and Stevens [18] in the same set of experiments in which the ligand hyperfine interaction was first observed, and an explanation given by Stevens 1151. Effectively, the reduction is caused by the transfer of d electrons to the ligand orbitals by covalency, and the orbital reduction factor is defined :

where I is the orbital angular momentum operator and

1

d

>

and

1

$

>

are free ion d orbitals and molecular orbitals. Orbital reduction factors have been reviewed with reference to EPR [20, 331 and magnetic susceptibility measurements [38, 391. For interactions within t2, .n orbitals

k,,

= 1

-

2 f , (30) and between n and rs orbitals

1

k,, = 1

-

- 2 N , N , x

For 1rC1;- the orbital reduction factor [20] gave

f , = 8

%

via eq. (30), in quite good agreement

with the value from the LHFI, although the g-value determination was later revised upwards to 14

%

in a more comprehensive analysis [35] involving the effect of excited and charge transfer states. For ions (e. g. dZ, d7 in an octahedral field) whose ground states are mixtures of e, and t,, configurations both k,,

and kc, appear in the overall orbital reduction factor and the analysis [34] for COF:- has already been mentioned.

(10)

COVALENCY EFFECTS IN MAGNETIC INTERACTIONS C6-547 8 h in electronic spectra might be expected to show a Ag = 2.0023 -

-

A (32) decrease

was less than expected for the free ions which was explained by Owen [40] in terms of covalency, and this evidence for covalency in transition metal com- plexes has also been widely quoted. A is the ligand field splitting and for the free ion I =

+

c/2 S, where ( is the spin orbit interaction for one electron. The treatment is quite analogous to the derivation of the orbital reduction factor and was reviewed by Owen and Thornley [20], and calculations performed in detail for N~F;- by Misetich and Wat- son [41]. More recently, for V F ~ - , it was shown [42] in a similar treatment that the observed g-factor was consistent with the spin transfer coefficients determined from the LHFI. In the covalent situation

. _where_{k,, is the orbital reduction factor, and}

where

5,

is the spin-orbit coupling constant for the central d orbit and

5,

that for the ligand p orbit. In addition to taking into account the covalently induced spin-orbit coupling on the ligand it is also necessary to consider effects due to coupling with charge transfer states. Such a complete analysis has only apparently been carried through for the ~ i " and V2+ fluoride complexes, and even here, as is the case with the orbital reduction factor, the fairly complex relation describing Ag makes it most useful as a check on covalency parameters determined by measurement of LHFI. It does appear, however, that the analyses carried out do not demand

a

significant expansion of the d electron radial functions, although is was originally suggested [43] that such an effect might explain the observed results.

For the most part spin-orbit effects have been interpreted in terms of an apparent reduction of A (eq. (32)) where Ag,,, = 2.0023

-

8 l'/A. A table of X/A, was given [20] for d3 and d8 ions (Table I). In view of the several factors entering into A', which will not necessarily change similarly from ligand to ligand such data gives a very qualitative estimate of the total covalency and relative magnitudes may not be particularly significant.

Some parameters available from electronic spectroscopy have been discussed recently 1131. Ligand field splittings are broadly interpretable in terms of the nature of the bonding in a similar fashion to partial isomer shifts 1441 and partial quadrupole splittings [45], although it is difficult to understand some features of the spectrochemical series, such .as the order of the halide ions. The nephelauxetic series, derived from the Coulombic repulsion terms

Spin-orbit coupling reductions (A'/&) ,for d3 and d8 ions

from free ion values according to N4 where N is the normalisation factor (eqs. (6), (7)) and also according to changes in < r-'

>

for the d orbitals. The nephelauxetic series for Iigands [I] :

I- > Br- > CN- 2 C1- > OH- >

is indeed fairly similar, for example to a table [I] of enthalpy discrepancies between compounds of the similar-sized Ca2+ and Cd2+ :

> NO, >

so:-

> F- ₍₃₆₎

and the nephelauxetic series for metal ions [I] :

roughly follows expectations of covalency. Also the effect is much weaker for rare earth ion f-f spectra than for transition metal spectra. The detailed mechanism underlying such series do not seem to be very well understood, however, and a recent calculation [46] for N~F:- perhaps encourages caution in drawing detailed conclusions on d-electron distributions from the effect.

The hyperfine field at the metal nucleus, observed particularly extensively for the d5 high spin ions Cr+, MnZf and Fe3+ by EPR and by the Mossbauer effect has long been known [47] to be reduced as the covalency of the metal-ligand bond increases (according to the electronegativity difference for example) (Fig. 3). In contrast to the hyperfine fields at ligand nuclei discussed above however, where the core polarisation term is small, for metal ions containing unpaired electrons this term is often dominant and, because of the several factors which can contribute, a detailed interpretation was not forthcoming for a long time. Spin polarised covalently occupied outer 4s orbitals [48]

(11)

C6-548 B. C. TOFIELD

FIG. 3.

-

The hyperfine interaction constant of Mn2+ plotted as a function of a covalency parameter, c, divided by the number of ligands, n. c is derived from the electronegativity difference of Mn and the ligand using the expression of Hannay and Smyth

(from ref. [48]).

reduces the s orbital polarisation) must all be taken into account. Hyperfine interactions and the Moss- bauer isomer shift were comprehensively discussed [49] by Sawatzky and van der Woude recently and we summarise their discussion below.

First of all however, let us mention metal core level splittings observed by photoelectron spectroscopy of transition metal compounds as these should directly reflect the magnitude of the spin and thus covalency effects. The splitting of 3s levels is the result of exchange coupling between the remaining 3s electron and the unpaired 3d electrons and although the magnitude is influenced by other effects also the order should correspond to the d electron covalency (as reflected by the total metal spin).

For Mn2 + the order is

MnF, :6.5eV(ref [50]), 6.3eV(ref [51]), MnCl, : 6.0 eV (ref [51]), MnO : 5.7 eV (ref [50]),

5.5 eV (ref [51]), MnS : 5.3 eV (ref [51]), MnBr, : 4.8 eV (ref [51])

.

₍₃₈₎

This series is in agreement with most other data which indicate fluorides to be the most ionic salts but it is not clear how the overall order should be interpreted.

For Cr3 + some splittings are [51,52] :

CrF, : 4.2 eV, Cr203 : 4.1 eV, CrC1, : 3.8eV, K,Cr(CN), : 3.6 eV and CrBr, : 3.1 eV

.

(39)

In particular we note the location of the cyanide complex. The initial measurement [51] found no splitting for this compound, but is was recently shown [52] that this was probably the result of impu- rity effects. It was pointed out [52] that the splitting of 3.6 eV is in accordance with the hyperfine interactions at the Cr nucleus which are typical of Cr3' compounds. The discussion [49] of the Mossbauer isomer shift, hyperfine and supertransferred hyperfine interactions has rather elegantly related these three effects for high-spin Fe3' oxides in particular and also permitted quantitative contact with covalency parameters determined by neutron diffraction. The discussion was pursued via an LCAO molecular orbital model with all the effects of covalency and overlap being incorporated into the ligand orbitals, i. e. the bonding orbitals, instead of primarily via the antibonding orbitals (eq. (1)-(5)) normally used to interpret LHFI and neutron diffraction. It is helpful to follow this interpretation. The metal wavefunctions in this approach are unchanged. We note that the transfer parameters b of ref. [49] are synonymous with y (eq. (9)) and B, is related to y as .f is to A.

A2 is synonymous with f (this was a convention adopted by Hubbard and Marshall [53] in discussing the effects of covalency in neutron scattering).

The problem then is to calculate the charge density at the nucleus, ]$(0)

12,

to calibrate the isomer shift, and the spin density at the nucleus to calibrate the hyperfine field. The isomer shift (IS) may be written

where ARIR is the fractional nuclear radius change on excitation, not known from other studies, cc is the calibration constant for the IS, and S 1$(0)

l2

is the change in charge density at the nucleus from source to absorber. The hyperfine interaction Hamil- tonian is given by eq. (15). The contact field at the nucleus is from ey. (15) and (23)

where 1$,(0)

l2

is the spin density at the nucleus. Both IS and Hco, involve only s orbitals so, separating up and down spin orbitals, for the isomer shift

is needed and for the hyperfine field

For high spin Fe3+ compounds other contributions to the hyperfine field may be neglected.

(12)

COVALENCY EFFECTS I N MAGNETIC INTERACTIONS C6-549

lumped into the ligand wavefunction, this has the

We note that the covalency parameter b is inversely proportional to the energy difference between the ground, and excited states with one electron transferred from ligands to metal, and a dependence on the electronegativity difference is observed for the hyperfine field as mentioned above (Fig. 3) and also, where other factors are constant, for the isomer shift (Fig. 4). For these high spin iron halides the decreasing IS with decreasing ligand electronegativity indicates that ligand-to-metal 4s donation is probably dominant. However an opposite trend is observed for example for low spin divalent iron dichalcogenides (F. W. D. Woodhams, personal communication) and in such a case ligand-to-metal 3d bonding is presu- mably giving the larger effect.

general form : 1.4

I

= N i

(

Ii

-

S S i j C j

+

Z:bi.q;

where $, is the free ion ligand wavefunction, p j a _1.2 filled metal wavefunction and q i an empty metal

wavefunction. The second term describes the orthogon- \ m

alisation and the third the covalency. The normalisa-

_-

E _E

tion constant is given by ₄

1.0- U

FIG. 4. -The isomer shift of divalent iron halides plotted against the average electronegativity of the ligands (fromref. 149491).

- FeSiF6.6H 0 ' ~ e ~ x FeCt2+LH20 FeCI2-2H20 - FeCI2-H20 FeCI, FeBr2 Fe12

Using wavefunctions of this type with the metal functions unchanged, parameters such as those of interest here are then the sum of the metal free ion term and the contribution arising from (eq. (42)). For the charge and spin densities at the nucleus where only the ligand a,, p orbital combination is of the appropriate symmetry, and the only metal orbitals directly involved are the 1s to 3s filled core orbitals and the initially empty 4s orbitals, then we have :

and

The first terms are the free ion values and the second _an3*₌

_anid

₊

_anid

₌₂_~ _;

_{~ , 2}

₂ ₊ terms have an overlap distortion contribution from

the orthogonalisation procedure and a 4s covalency

+

3NL2 S:

+

2 ~ ; ' b , 2

+

3 N i 2 b , 2 . (47) term. smaller correction terms involving ligand

orbitals have been omitted. In oxides at least the first terms are dominant in both but will scale according to the d electron populations.

For the isomer shift this can be expressed as :

Thus comparing eqs (44) and (46), (47) the opposing effects of ligand to-metal transfer to 3d and 4s orbitals are seen. On the other hand metal-to-ligand back- handing will decrease the 3d shielding and for 57Fe, decrease the IS. The similarity of the IS of F ~ " ( c N ) ~ -

3

and F~"'(CN);- was interpreted 1541 as arising from 2

C

I

cpns(0)

l2

= C

-

1-8(an3d)

n = 1 (46) the transfer of the extra ferrous ion electron on to the

ligands by n back-bonding.

where C is the free ion density at the nucleus and If the 3d or 4s orbital populations are known an3, is the change in the total d electron population. or can be calculated, the calibration constant a The constant of proportionality was fixed [49] from of eq. (40) can be determined. This would make the Hartree-Fock calculations for the free ions Fe3+ isomer shift a very useful parameter for investigating

(13)

C6-550 B. C . TOFIELD

states where there is a significant spread of values, and especially in conjunction with d orbital population evidence from other measurements. Nevertheless, although there has been much effort directed towards this end in the last few years, the several contributing influences, as exemplified in the examples mentioned, will encourage circumspection in drawing detailed conclusions about bonding.

In suitable circumstances both 3d and 4s occupations may be extracted from hyperfine field data as discussed by Sawatzky and van der Woude [49]. This was done for a few different situations but most elegantly for the rare earth orthoferrites [55] where, in addition, comparison of the d orbital population could be made with values from neutron diffraction data determined [56] by myself a few years ago. The procedure adopted is one which allows the determination of the bond-angle independent portion of the supertransferred hyperfine field, STHF, part of the total hyperfine field. The 3d and 4s orbital populations can then be extracted using catalogued wavefunctions.

The first term of eq. (45) will be scaled by the change in spin of the 3d electrons arising from overlap and covalency. For Fe3+, with the up spin d states full and the down states empty, the up spin occupation will decrease due to charge transfer and increase due to overlap, with the opposite being the case for the down spin occupation. The net change in spin is the difference of the changes in up and down spin populations :

central cation s orbitals by ligand orbitals unpaired by transfer into unoccupied 3d orbitals of the neighbouring cations. The latter is dominated by p to d transfer and for antiferromagnets such as the rare earth orthoferrites, where one spin state is full and the other empty of d electrons, the orbitals on the ligands relatively depopulated by transfer to the outer cations will be of up spin if these Fe3+ ions have down spin. This means that the overlap distortion of the central Fe3' down spin s orbitals will be slightly greater than of the up spin orbitals, so that the down spin density at the central cation nucleus will be the

greater.

The reduction in the negative, up spin overlap distortion part of the second term of eq. (45), which gives rise to the STHF is determined by the bonds from the ligands to the surrounding cations and may be expected [55] to depend on spin transfer in a similar kind of way to superexchange. We will not discuss the relation between covalency, exchange interactions, and magnetic ordering temperatures in any detail here because, although aspects of this relation are quite clear, the process can only be to rationalise qualitatively magnetic ordering temperatures using known covalency parameters and not the other way round. The subject was well discussed by Owen and Thornley [20], and more recently, for example by Sawatzky et a1 1571 who also consider the STHF and superexchange in an ion like Cr3+ where the e, metal orbitals are initially empty but exchange split. We will just note for reference, however, that the magnetic ordering temperatures of KMnF,, KCOF, and KNiF, (88 K, 125 K and 275 K respectively), scale well with the covalency parameters determined by spin resonance and neutron diffraction [20]. For these perovskites an 1800 interaction

- 2 Nb2 bb2 - 3 3:' b121 (48) _{will be dominant.}

where Boekema et al. [55] treated -the STHF in the ortho-

ferrites by reference to the superexchange reflected NJ,, = (1

-

s $ , J - ~ / ~

_{in the magnetic ordering temperatures, noting that}

N:,, = (1

+

2 b,,, So,,

+

b:,,)a112. (49) Jiuo

a f ~ f ~

(neglecting bonding) and J90 a 2

f&.

If the magnetic ordering temperature, TN, is pro- Thus the free ion contribution to the hyperfine portional to J, thenin this approximation

field is

Hfree [(So

-

Ahy)/.yo]

where H,,,, has been calculated to be - 630 kOe. The second term in (eq. (45)) comprises contributions from overlap distortion of the iron core orbitals and the differential charge transfer to the spin polarised 4s orbitals, of which the latter is dominant [49] and would give a field of

-

420 kOe for a full 4s shell. Finally, in concentrated magnetic materials such as the orthoferrites there is a small correction to the second term of eq. (45). This additional, supertransferred, hyperfine field (STHF) arises from the neighbouring Fe3 + ions. For neighbouring ions

with spin, a supertransferred hyperfine field at a central ion arises from ov'erlap distortion of the

(14)

COVALENCY EFFECTS M MAGNETIC INTERACTIONS C6-551

in 8 bringing a second oxygen o orbital more into play.

Writing

TN

= TN (900) +

(

TN (1800) - TN (900)

)

x

x c0s2 8 (51)

extrapolation of the observed linear curve of TN

against cos2 8 gave [55] f,

If,

= 5.3, in reasonable agreement with 4from neutron and LHFI data (below). The general expression for the STHF correction is obtained by considering ligand a orbital overlap with the central cation s orbitals, the neglect of .n

interaction being justified by the data just discussed. Because 8 # 1800 the ligand orbital required will be a mixture of the o and n bonding orbitals used to bond to the outer Fe3+

p, = p, cos 8

-

p, sin 8 ₍₅₂₎

and the supertransferred spin density at the central Fe3 + nucleus takes the form,

where I is the number of nearest neighbour irons. Thus there are three contributions to the observed hyperfine field :

where HA,, is H,,,,(S,

-

AS)/&, Hco, arises from the second terms of eq. (45) and HsTHF comes from eq. (53). The angular dependent term involved in HsTHF may be rewritten as

{

(f, - f,) cos2 8

+

f,

}

and thus this treatment predicts a linear variation also of H, as well as of TN, with cos2 6' which is indeed observed (Fig. 5). Using the relative magnitudes of fa and f, determined

from the Ntel temperature variation an angular independent part of HsTHF involving only f, in eq. (53) could be determined from this plot and thus the magnitude of (~f',,,

+ Hco,) found as well. As already

mentioned H,,,, and Hco, had been calculated for a fully ionic Fe" ion (- 630 kOe) and for a full 4s shell (420 kOe) so that, using free ion wave functions and an average Fe-0 distance of 2.01 1

A,

the various overlap integrals and normalisation constants were determined and the 3d and 4s covalency parameters scaled to fit the measured fields. The values found were [55] f, = 0.144, f, = 0.027 and 4s occupation 0.10 electrons. .fa and f, are somewhat higher than found from neutron diffraction and spin resonance work (see below).

In view of the difficulties involved in extracting such numbers from the Mossbauer data such agreement must be viewed as very encouraging. Although the orthoferrite series is fairly ideal for such a study the angular range which is accessible (- 150 in total) is quite limited and it would also be interesting to know if the effect of direct metal-to-metal transfer on the

FIG. 5. - The hyperfine field at 57Fe measured by the M O S ~

bauer effect for the rare earth orthoferrites plotted against

( C O S ~ where 0 is the Fe-0-Fe bond angle. This plot was

used to derive information on the supertransferred hyperline field and thus on the covalency parameters of the ~ e 0 2 - clusters

(from ref. [49]).

estimated STHF has been estimated. It would appear that in some cases [58] such a direct transfer may be significant. Nevertheless, if 57Fe Mossbauer data can be used to provide quantitative covalency data in a complementary manner to neutron diffraction and LHFI this is a significant step forward although a thorough analysis as was carried out for the orthoferrites will always be needed. For example, it has been shown recently that the isomorphous replacement 2 02- = F -

+

N3- may sometimes be made. It has not been possible to study nitrides at all by the other methods mentioned here, although one presumes the covalency order will be F- < 02-

<

N3-. But Mossbauer data of the high spin Fe3+ fluoronitride Fe,N,F3, which has the cubic bixybite (Fe, Mn),03, structure with apparently random F- and N3- ions have recently been published 1591 and the hyperfine fields at the two metal sites (505 kOe and 475 kOe) are much less than observed for Fe3+ fluorides. Deter- mination of the covalency for F - and, particularly, for N3- by the methods used for the Fe3' oxides would be very encouraging. A relatively new method of determining STHF uses the Perturbed Angular Correla- tion of y-rays (PAC) technique, described [60] recently for lllCdm in fluorides containing transition metal ions. So far, the analysis does not seem te have proceeded sufficiently to achieve consistency with neutron diffraction and spin resonance data.

(15)

C6-552 B . C . TOFIELD

fine field, an analysis is more complicated than for the normalised to unity. The form factor of the magnetic ligand quadrupole interactions discussed above, moment distribution as expressed in eq. (13) gives, although in principle, such interactions should reflect at Q = 0,

the central field covalency through the

<

r - 3

>

term.

For divalent iron compounds, where the valence efg will f,(o) = 1

-

n,2

-

a,"

be present in first order, partial quadrupole

splittings [45] seem to provide analogous qualitative fb(0) = 0 (510 information to partial isomer shifts 1441 and the spectro- - . f,(o) = 2:

+

A:

chemical series, as mentioned above. However, because in any particular compound, the quadrupole splitting depends on the crystal field splitting and the spin orbit coupling as well as on central field and charge transfer covalency, a detailed analysis to provide information on either of the latter effects does not seem to be easily achieved at the present time [61, 13, 491.

5. Neutron diffraction investigations.

-

Boekema et al. [55], in discussing the comparison of covalency parameters determined from Mossbauer data with neutron values indicated that the expression used in determining covalency parameters from moment reductions observed by neutron diffraction was obtained by assuming 2 = y (eq. (9)). The first terms of eqs. (13) or (14) used to estimate A or f from moment reductions observed by neutron scattering, certainly do not provide the same moment reduction as given by Boekema et al. (eq. 48). eqs. (13) and (14) may however be rearranged to :

I+,,(r)

l2

= d:(1

+

2 A, S,

+

2 As Ss

-

2: - A,")

-

By putting b = y = A - S and neglecting spin polarisation we see that eq. (48) and the first terms of eqs. (55) and (56) are now in agreement, and that the true moment reduction associated with the metal d function is indeed somewhat less than given by the (1 - A2) term of eqs. (13a) and (14a), because of the effect of overlap terms in AS. When considering the effective magnetic moment observed in the scattering from an antiferromagnet such as LaFeO,, however, the spatial extent of the magnetic moment distribution, which is reflected in the angular dependence of the scattered intensity, must be considered.

This is given by the magnetic form factor, which in the spin only case may be simply written as

where D(r) is the normalised spin density, Q is the scattering vector

(1

Q

I

= 4 n sin O/A), and f (Q) is

and analogously for eq. (14). In other words, the only contribution to magnetic scattering at Q = 0 comes from the first terms of eqs. (13) and (14) containing no overlap terms of the form AS, and the ligand spin term. Moreover, in antiferromagnets of the orthoferrite type where each Iigand is symmetrically surrounded by metals with opposite spin (shown in one dimension - .

in figure 6) the l&nd moment disappears so that only

FIG. 6 . - A diagram showing the cancelling of the covalently induced ligand moment in a simple antiferrornagnet. For clarity

this is shown in one dimension only.

the metal term in fact is present at Q = 0. The various overlap terms represented in the second term of eq. (13) peak at sin 0/A

-

0.2-0.3. As the magnetic intensities used to determine accurate magnetic moments have generally been measured at sin 011

-

0.1 it has been assumed that this overlap term will be small and that to a good approximation the observed moment reduction is given by

A

rather than by A(l

-

S ) . Of course the product of the spin and form factor is measured at any particular' Q and, in the absence of other information concerning the detailed shape of the form factor, the free ion shape has been assumed. This is given by the dotted line in figure 7 where the different contributions to the form factor are detailed. The moment reductions expected for high spin ions in octahedral coordination are given in Table 11, together with covalency parameters determined from the LHFI for spin only ions.

(16)

COVALENCY EFFECTS IN MAGNETIC INTERACTIONS C6-553

FIG. 7. - A schematic representation showing the effect of covalency on the magnetic form factor according to the simple molecular orbital theory of ref. [53]. The metal, overlap and ligand contributions according to eqs. (13) and (58) are shown as dotted lines, and the combined paramagnetic form factor is also indicated. The metal spin form factor has the same shape as the free ion form factor and for convenience in this diagram the effect of the covalent moment reduction is shown as a reduction in the magnitude of the form factor (in the text the usual convention is followed that f ( Q ) is always normalised to unity). In an antiferromagnet the ligand moment is quenched, but above

Q % 20 nm-1 the antiferromagnetic and paramagnetic form factors are very similar. The covalent form factor is slightly expanded relative to the metal curve because of the effect of overlap, but at Q x 12 nm-1 (sin 0/1

-

0.1) where magnetic reflections are generally measured to estimate the magnetic moment, the assumption of the free ion form factor is a fairly

good approximation.

moment observed were subsequently explained by Hubbard and Marshall [53] who first presented the molecular orbital interpretation of the effects of covalency in neutron scattering reproduced here. Quite good agreement was obtained [53] with the experimental curve when the effects of the small orbital moment and the spin polarisation of the inner shells were included.

More recently, the magnetic moment of Ni2' in NiO was measured by Fender et al. [64] using a polycrystalline sample by comparison of the first magnetic reflection ($*$) with adjacent nuclear reflections [(I 11) and [2OO)J. The use of the real form factor for this compound, measured by Alperin [63] and the calculated 1621 free ion form factor provided an opportu- nity to assess the error involved in the normal approximation. The actual value of the form factor at the magnetic

(444)

reflection is 0.937 whereas the free ion calculated value is 0.914. The observed moments will scale inversely (S,/S, = f,lfi) and thus will differ by less than 3

%.

However, because the covalency para-

TABLE II

Moment reductions for antiferromagnets (octahedral complex)

Covalency

Number of parameter

electrons Moment reduction from LHFI

-

1 ( g e ; 4 fn ---

2 (t;g e,O) 4fn ---

3 (ti, e;) 4fn f z

meters are obtained from the quite small difference between the measured and the expected spin, this small effect on the observed moment has a much larger effect on the covalency parameters, which change from

fu

+

f,

= 3.8(2)

%

using the Alperin form factor to

3.1(2)

%

using the free ion form factor, a difference of 18

%.

Even this difference, however, which reflects a part of the negative overlap contribution to the moment reduction (eq. (55a)) is not qualitatively significant. When we come to examine the data for Fe3+, although the moment reduction given in ref. [56] is increased slightly when a revised value for the scattering length of germanium (which was used to calibrate the intensity of the first magnetic reflection (orthorhombic (001)+ (101) at sin O/il=O.ll)) is used (0.8193 (ref. [65]) c. f. 0.84) the new value off,

+

2 fn(l 1.4 $. 1.2

%)

[27] is considerably less than the value found by Boekema et al. (20

%)

[55]. As we have shown, the form factor

at Q = 0 is reduced by the magnitude of il only, not including overlap, and at sin 012

-

0.1 a small part of the overlap moment of eqs. (13b) or (14b) will apply as discussed for NiO. This term itself is somewhat smaller, because of the negative two-centre overlap terms, than the adjustment needed to bring (l3a) or (14a) into line with eqs. (48), and (55a) and (56a), so that a correction similar to that found for NiO might be expected (bringing f,